CC1 Chain Rule

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From users who are members of Skills Audits for Maths and Stats :

Will Morgan	said	Ready to use	1 year, 3 months ago
Mash Sheffield	said	Needs to be tested	1 year, 5 months ago

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Will Morgan 1 year, 1 month ago

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Will Morgan 1 year, 3 months ago

Gave some feedback: Ready to use

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Mash Sheffield 1 year, 5 months ago

Gave some feedback: Needs to be tested

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Mash Sheffield 1 year, 5 months ago

Created this as a copy of Differentiation: Chain Rule 4a.

Name	Status	Author	Last Modified
Differentiation: Chain Rule 4a	Ready to use	Ben McGovern	16/12/2022 10:30
Differentiation: Chain Rule 5a	Ready to use	Ben McGovern	22/09/2022 15:09
Differentiation: Chain Rule 12	Ready to use	Ben McGovern	22/09/2022 15:38
Differentiation (Harder powers - negative and fractions) 5	draft	Ben McGovern	27/09/2022 09:01
CC1 Chain Rule	Ready to use	Mash Sheffield	17/04/2025 09:02

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Question statement

Calculate the derivative of $y=\simplify[all]{sin({a}*x^{n}+{b}*x^{m})}$ .

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			Name	Type	Generated Value

a	integer	4
b	integer	1
n	integer	5
m	integer	4

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1. Gap Gap 0 Mathematical expression
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$\frac{dy}{dx}=$ Gap 0

Advice

If we have a function of the form $y=f(g(x))$ , sometimes described as a function of a function, to calculate its derivative we need to use the chain rule:

$\frac{dy}{dx} = \frac{du}{dx} \times \frac{dy}{du}.$

This can be split up into steps:

Let $u=g(x)$ ;
Rewrite $y$ in terms of $u$ , such that $y=f(u)$ ;
Calculate $\frac{du}{dx}$ and $\frac{dy}{du}$ ;
Write $\frac{dy}{dx}$ as a product of $\frac{du}{dx}$ and $\frac{dy}{du}$ ;
Make sure $\frac{dy}{dx}$ is only in terms of $x$ . Ensure any $u$ terms have been replaced using the initial substitution.

Following this process, we must first identify $g(x)$ . Since the function is of the form $y=f(g(x))$ , we are looking for the 'inner' function.

So, for $y=\simplify[all,fractionNumbers]{sin({a}*x^{n}+{b}*x^{m})}$ ,

$g(x)=\simplify[all, fractionNumbers, unitFactor]{{a}*x^{n}+{b}*x^{m}}.$

If we now set $u=g(x)$ , we can rewrite $y$ in terms of $u$ such that $y=f(u)$ :

$y=\simplify[all, fractionNumbers,unitFactor]{sin(u)}.$

Next, we calculate the two derivatives $\frac{du}{dx}$ and $\frac{dy}{du}$ :

$\frac{du}{dx}=\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}}, \quad \frac{dy}{du}=\simplify[all, fractionNumbers, unitFactor]{cos(u)}.$

Plugging these into the chain rule:

$\begin{split} \frac{dy}{dx} &= \frac{du}{dx} \times \frac{dy}{du}, \\&=(\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}}) \times\simplify[all, fractionNumbers, unitFactor]{cos(u)}. \end{split}$

Finally, we need to express $\frac{dy}{dx}$ only in terms of $x$ , so we must replace the $u$ term using the initial substitution $u=\simplify[all, fractionNumbers, unitFactor]{{a}*x^{n}+{b}*x^{m}}$ :

$\frac{dy}{dx} =(\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}})\simplify[all, fractionNumbers, unitFactor]{cos({a}*x^{n}+{b}*x^{m})}.$

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Will Morgan 1 year, 1 month ago

Will Morgan 1 year, 3 months ago

Mash Sheffield 1 year, 5 months ago

Mash Sheffield 1 year, 5 months ago

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CC1 Chain Rule

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Will Morgan 1 year, 1 month ago

Will Morgan 1 year, 3 months ago

Mash Sheffield 1 year, 5 months ago

Mash Sheffield 1 year, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`